Linear transvection groups (Algebraic Combinatorics)

Linear

transvection

groups

Hans

Cuypers

Anja Steinbach

1

Introduction

Most classical groups arising from (anti-) hermitian forms or (pseudo-) quadratic

forms contain so-called isotropic transvections. Indeed, suppose, for example, that $V$

is a vector space over some skew field If endowed with a $(\sigma, -1)$-hermitian form $f$

and let $w$ be an isotropic vector of $V$, i.e., $f(w, w)=0$, with $w\not\in \mathrm{R}\mathrm{a}\mathrm{d}(V, f)$

.

Then,

for each $a\in I\{’\mathrm{W}\mathrm{i}*\mathrm{t}\mathrm{h}a^{\sigma}=a$, the map _{$v\vdash\Rightarrow v+f(v, w)aw$} for $v\in V$ is a transvection

fixing the form $f$

.

The isotropic transvection subgroups of these classical groups, i.e., the subgroups

generated by all isotropic transvections with a fixed axis, form a class $\Sigma$ of abelian

subgroups which is a class of abstract transvection groups in the sense ofTimmesfeld

[25]. This means that for all $A,$$B\in\Sigma$ we have that $[A, B]=1$ or $\langle A, B\rangle$ is a rank 1

group (i.e., $A\neq B$, and for each $a\in A\#$, there exists some $b\in B\#$ with $A^{b}=B^{a}$).

Here we describe a common characterization of all these classical groups with

isotropictransvections as lineargroups generated by a class $\Sigma$ of abstract transvection

subgroups such that the elements of $A\in\Sigma$ act as transvections.

Details and proofs of the results mentioned in this paper can be found in [8] and

will appear elsewhere.

2

$\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}$

2.1 Notation. Suppose $V$ is a left vector space of arbitrary dimension defined over

some skew field $K$

.

For any linear map _{$t:Varrow V$} (acting from the right) and vector

$v\in V$, we define the commutator $[v,t]$ to equal vt–v. Here $vt$ is the image of $v$

under the linear map $t$

.

If $W$ is a subspace of $V$ and $S$ a set of linear maps of$V$, then

$[W, S]$ is the subspace of $V$ spanned by _{$\{[w, s]|w\in W, s\in S\}$}

.

An invertible linear map $t:Varrow V$ is called a transvection if

(b) [V, $t$] $\subseteq C_{V}(t)=\{v\in V|vt=v\}$.

Suppose $t:Varrow V$ is a transvection. From the definition it is clear that $C_{V}(t)$ is a

hyperplane of $V$, it is called the axis of $t$

.

The 1-dimensional subspace [V,$t$] is called

the center of $t$

.

Let $v_{t}$ be a vector spanning the center [V,$t$] of$t$

.

Then there is an element$\varphi\in V^{*}$,

the dual of $V$, with kernel $C_{V}(t)$ such that the action of $t$ on $V$ can be described as

follows:

$t$ : _{$v\vdash+v+(v\varphi)v_{t}$} for $v\in V$

.

2.2 Transvections in classical groups. In this paper we consider subgroups

of the general linear group on $V$ which are generated by transvections. For

finite-dimensional $V$, it is well known that the special linear group $\mathrm{S}\mathrm{L}(V)$ is generated by

its transvections. For infinite-dimensional $V$, the subgroup of $\mathrm{G}\mathrm{L}(V)$ generated by

the transvections is finitary, i.e., for each element$g$ of this subgroup the commutator

[V,$g$] is finite-dimensional. In fact the transvections generate the full finitary special

linear group $\mathrm{F}\mathrm{S}\mathrm{L}(V)$.

Also the classical subgroups of $(\mathrm{F})\mathrm{S}\mathrm{L}(V)$ arising from (anti-) hermitian forms

contain transvections. Indeed, suppose $V$ is endowed with a $(\sigma, -1)$-hermitian form

$f$ and let $w$ be an isotropic vector of $V$, i.e., $f(w, w)=0$, with $w\not\in \mathrm{R}\mathrm{a}\mathrm{d}(V, f)$. Then,

for each $a\in K^{*}$ with $a^{\sigma}=a$, the map

$t:v\vdash+v+f(v, w)aw$ for $v\in V$

is a transvection fixing the form $f$

.

Such a transvection will be called isotropic with

respect to $f$, cf. Hahn and O’Meara [10, p. 213]. If the dimension of $V$ is finite, then

the subgroup of $\mathrm{S}\mathrm{L}(V)$ of isometries of $f$ is generated by its isotropic transvections.

Similarly, for infinite-dimensional $V$, the isotropic transvections leaving $f$ invariant

generate the finitary subgroup of the corresponding classical group.

Maybe less well-known is the following class of transvections which we find in

orthogonal groups. Suppose $\sigma$ is an involutory anti-automorphism of $I${: and for

$\epsilon\in\{-1,1\}$, set A $:=\{c-\epsilon c^{\sigma}|c\in I\acute{\iota}\}$

.

Now consider a non-degenerate

pseudo-quadratic form $q$

:

$Varrow I\iota^{\nearrow}/\Lambda$ with associated trace-valued $(\sigma, \epsilon)$-hermitian form

$f$

:

$V\cross Varrow K$, see Tits [26, (8.2.1)] (a radical of$f$ is allowed). Let $w$ be an isotropic

vector of $V$, i.e., _{$q(w)=0+\Lambda$}

.

If there exist $a\in K^{*}$ and $r_{a}\in.\mathrm{R}\mathrm{a}\mathrm{d}(V, f)$ (possibly $0$)

with $q(r_{a})=a+\Lambda$, then the map

is a transvection in the isometry group of $q$, which we also call an isotropic

transvec-tion. The axis of $t$ is the space $w^{\perp}=\{v\in V|f(v, w)=0\}$, its center is $\langle aw+r_{a}\rangle$.

Such transvections exist provided that $q$is not an ordinary quadratic form with trivial

radical Rad$(V, f)$.

We notice that these isotropic transvections act trivially on Rad$(V, f)$ and

there-fore also induce transvections on the space $V/\mathrm{R}\mathrm{a}\mathrm{d}(V, f)$

.

2.3

Transvection

subgroups. Let $t_{1}$ and $t_{2}$ be two transvections on $V$

.

Up to

symmetry we only have the following three possibilities for the centers and axes of$t_{1}$

and $t_{2}$:

(1) [V,$t_{1}$] $\subseteq C_{V}(t_{2})$ and [V,$t_{2}$] $\subseteq C_{V}(t_{1})$, then $[t_{1}, t_{2}]=1$,

(2) [V,$t_{1}$] $\not\subset C_{V}(t_{2})$ and [V, $t_{2}$] $\not\subset C_{V}(t_{1})$, then $\langle t_{1}, t_{2}\rangle$ is contained in the group

$\mathrm{S}\mathrm{L}([V, t1]\oplus[V, t_{2}])\simeq \mathrm{S}\mathrm{L}_{2}(K)$,

(3) [V,$t_{1}$] $\subseteq C_{V}(t_{2})$ and [V,$t_{2}$] $\not\subset C_{V}(t_{1})$, then $[t_{1}, t_{2}]$ is also a transvection on $V$

with center [V,$t_{1}$] and axis $C_{V}(t_{2})$

.

If $t_{1}$ and $t_{2}$ are isotropic transvections with respect to some anti-hermitian form $f$,

or some pseudo-quadratic form $q$ with associated $(\sigma, \epsilon)$-hermitian form $f$, then case

(3) does not occur. Indeed, for all $v,$$w\in V$ we have that $f(v, w)=0$ if and only if

$f(w, v)=0$.

Now suppose $t_{1}$ and $t_{2}$ are two isotropic transvections with respect to some

anti-hermitian form $f$ or pseudo-quadratic form $q$. Denote by $T_{1}$ and $T_{2}$, respectively, the

subgroup of $\mathrm{G}\mathrm{L}(V)$ generated by all isotropic transvections with the same axis as $t_{1}$

or $t_{2}$, respectively. These subgroups are called isotropic transvection subgroups and

are isomorphic to $(I\acute{\iota}^{\sigma}, +)$, if we consider the isotropic transvections with respect to

the anti-hermitianform, and to $(\triangle, +)$ in case they leave the pseudo-quadratic form $q$

invariant. Here $K^{\sigma}=\{a\in I\mathrm{t}^{\nearrow}|a^{\sigma}=a\}$ and $\triangle=\{a\in L|$ there exists $r_{a}\in \mathrm{R}\mathrm{a}\mathrm{d}(V, f)$

with $q(r_{a})=a+\Lambda\}$

.

It is straightforward to check that for $T_{1}$ and $T_{2}$ we have one of the following two

possibilities:

(1) [V,$T_{1}$] $\subseteq C_{V}(T_{2})$ and [V, $T_{2}$] $\subseteq C_{V}(T_{1})$, then $[T_{1}, \tau_{2}]=1$.

(2) [V,$T_{1}$] $\not\subset C_{V}(T_{2})$ and [V,$T_{2}$] $\not\subset C_{V}(T_{1})$, then $\langle T_{1}, T_{2}\rangle$ is isomorphic to the

sub-group

$\langle$

The subgroups $\mathrm{S}\mathrm{L}_{2}(K^{\sigma})$ and $\mathrm{S}\mathrm{L}_{2}(\triangle)$ of $\mathrm{S}\mathrm{L}_{2}(I\zeta)$ are rank 1 groups in the following sense, see Timmesfeld [25]: for each $x_{1}\in\tau_{1}\#$ there exists an $x_{2}\in\tau_{2}\#$ with $T_{2}^{x_{1}}=T_{1}^{x_{2}}$

.

Indeed, if $1\neq x_{1}=$, then for $x_{2}$ we may take

$K^{\sigma}$ or $\triangle$, respectively, then so is $\lambda^{-1}$

.

3

The

main

results

The main goalis togivea

common

characterizationof the various classical groups

(dif-ferent from the special linear group) as groups generated by their isotropic

transvec-tion subgroups. We now describe the exact setting we will work in:

3.1 Setting. Let If be a skew field and $V$ a vector space over $I\mathrm{t}’$

.

Assume that

$G\leq \mathrm{G}\mathrm{L}(V)$ such that:

(1) $G$ is generated by a conjugacy class $\Sigma$ of abelian subgroups of $G$.

(2) For $A,$ $B\in\Sigma$, either $[A, B]=1$ or $\langle A, B\rangle$ is a rank 1 group (i.e., $A\neq B$, and

for each $a\in A\#$, there exists some $b\in B\#$ with $A^{b}=B^{a}$).

(3) For $A\in\Sigma$, every $a\in A\#$ is a transvection on $V$.

(4) Each $A\in\Sigma$ contains at least 3 elements.

(5) There are $A,$$B\in\Sigma$ with $[A, B]=1$ and $c_{\Sigma}(A)\neq C_{\Sigma}(B)$.

(6) $V=[V, G]$

.

The conditions (1) and (2) on $\Sigma$ are the defining conditions of a class of abstract

transvection groups in a group $G$ in the sense of Timmesfeld [25]. In (5), we use

the definition $c_{\Sigma}(A)=\{T\in\Sigma|[A, T]=1\}$, for $A\in\Sigma$

.

By $P(V)$ we denote the

projective space corresponding to $V$

.

Notice that we do not assume that for each $A\in\Sigma$ the commutator space [V, $A$]

is 1-dimensional nor that $C_{V}(A)$ is a hyperplane in $V$

.

We are now able to state our first result:

3.2 Theorem. Assume $G$ is a subgroup

of

$\mathrm{G}\mathrm{L}(V)$ generated by a class $\Sigma$

of

abstract

transvection groups as in the setting above.

If

$C_{V}(G)=0$ ($e.g.,$ $V$ is irreducibfe), then $G$ is quasi-simple and we are in one

of

(a) There exist a skew

_field

$L$ with involutory anti-automorphism$\sigma$, some $\epsilon\in\{1, -1\}$

and a vector space $W$ over $L$ endowed with one

of

the folfowing

forms

(recall that

A $:=\{c-\epsilon c^{\sigma}|c\in L\})$:

(1) a non-degenerate pseudo-quadratic

_form

$q$ : $Warrow L/\Lambda$ with associated

trace-valued $(\sigma, \epsilon)$-hermitian

form

$f$ : $W\cross Warrow L$ or

(2) a non-degenerate $(\sigma, \epsilon)$-hermitian

form

$f$, where A coincides with

$\{c\in L|\epsilon c^{\sigma}=-c\}$

.

The group $G$ is isomorphic to the classical normal subgroup

of

the isometry group

of

$q$ or $f$ generated by the isotropic transvection subgroups $T_{\mathrm{p}}$, where

$p$ runs over the

points

_of

the classical polar space $Q$ arising

from

$W$ and $q$ or $f(i.e.,$ $p$ runs over the

isotropic points

_of

$P(W))$.

(b) There exists a quaternion skew

_field

$L$ (with standard $(anti-)involution\sigma$ and

center $Z(L))$

.

Denote by $W$ the vector space $L^{4}$ over $L$ endowed with the

pseudo-quadratic

_form

$q$ : $Warrow L/Z(L)$

defined

by $q(x_{1,2,3,4}xxX):=x_{1}x_{3}^{\sigma}+x_{2}x_{4}^{\sigma}+Z(L)$

for

$x_{1},$ $x_{2},$$x3,$$X_{4}\in L$ (with associated $(\sigma,$ $-1)$ hermitian$for^{j}mf$).

The group $G$ is isomorphic to the subgroup

of

the isometry group

of

$q$ generated by

those isotropic transvection subgroups $T_{p_{f}}$ where _$p$ runs over the points

of

a so-called

special subquadrangle $Q$

of

the classical generalized quadrangle arising

from

$W$ and $q$.

(c) There exists a non-perfect commutative

_field

$L$

of

characteristic 2 with

$L^{2}\subseteq\Theta’\subseteq L’\subseteq\Theta\subseteq L$,

where $L’$ is a

subfield of

$L,$ $\Theta$ is an $L’$-subspace

of

$L$ which generates $L$ as a ring

and $0’$ is an $L^{2}$-subspace

of

$L’$ which generates $L’$ as a ring. Denote by $W$ the vector

space $\Theta\cross(L’)^{4}$ over $L’$ endowed with the quadratic

form

$q$

:

$Warrow L’$

defined

by

$q(x_{0};(x_{1}, x_{2}, X3, x_{4})):=x_{0}^{2}+x_{1}x_{2}+x_{3}x_{4}$

for

$x_{0}\in\Theta$ and $x_{1},$ $x_{2},$$X_{3,4}x\in L’$ (with

associated symmetric bilinear

_form

$f$).

The group $G$ is isomorphic to the subgroup

of

the isometry group

of

$q$ generated by

those isotropic transvection subgroups $T_{\mathrm{p}_{J}}$ where _$p$ runs over the points

of

a so-called

mixed subquadrangle $Q$

of

the classical generalized quadrangle arising

from

$W$ and $q$.

In Cases (a) to (c), the elements

_of

$\Sigma$ are contained in the isotropic transvection

subgroups and in 1-1-correspondence with the points

_of

$Q$

.

The action

_of

$G$ on $V$ is induced by a semi-linear mapping $\varphi$ : $Warrow V$ (with

respect to an embedding $\alpha$ : $Larrow K$ resp. $\alpha$ : $L’arrow I\mathrm{t}^{\nearrow}in$ Case $(\mathrm{c}))$ with kernef

A detailed description of the generalized quadrangles in Cases (b) and (c) can be

found in Steinbach and Van Maldeghem [21].

In Theorem 3.2, we may conclude that any two transvections $t_{1}$ and $t_{2}$ in some

element $A\in\Sigma$ have the same center and the same axis. However, in the examples

that we have encountered in 2.2 we have seen that [V,$A$] need not always be a point

of $P(V)$. However, the condition $C_{V}(G)=0$ forces [V,$A$] to be 1-dimensional for all $A\in\Sigma$

.

Ifwe relax on the condition $C_{V}(G)=0$ but stillinsist on [V,$A$] being l-dimensional,

we find that most of the groups $G$ are classical groups, however, now defined by a

possibly degenerate form:

3.3 Theorem. Assume $G$ is a subgroup

of

$\mathrm{G}\mathrm{L}(V)$ generated by a cfass $\Sigma$

of

abstract

transvection groups as in the setting 3.1 above.

Assume in addition that [V,$A$] is 1-dimensional

for

all $A\in\Sigma$. Then there exists

a subspace $U$

of

$V$ and a subgroup $G_{0}=\langle\Sigma_{0}\rangle f\Sigma_{0}=\{A\in\Sigma|[V, A]\subseteq U\}$

of

$G$ such

that:

(1) $V=U+C_{V}(G)$ $($not necessarily a direct $sum)_{i}$

(2) $U$ and $G_{0}$ are as $V$ and $G$ in the conclusion

of

Theorem 3.2 with the possible

exception that the kernel

_of

the semi-linear mapping $\varphi$ is only contained in

Rad$(W, f)$

.

$Moreover_{f}$

if

$U$ and $G_{0}$ are as in Case (a)

of

3.2, then $G$ is isomorphic to the classical

group generated by the isotropic transvection subgroups on a vector space $\overline{W}=\underline{W}\oplus\tilde{R}$

over$L_{f}$ with extended

form

$q$ or$f$ such that

$\overline{R}$

is isotropic and contained in Rad$(W, f)$.

The action

_of

$G$ on $V$ is induced by a semi-linear mapping $\tilde{\varphi}:\overline{W}arrow V$ extending

$\varphi$

.

In the situation of Theorem 3.3, the classicalgroup generatedby the isotropic

transvec-tion subgroups on the vector space $\overline{W}=W\oplus\tilde{R}$is isomorphic to _{$(\oplus_{i\in I}W_{i})$}

:

$G_{0}$, where

each $W_{i}$ is a copy of the natural module $W$ for $G_{0}$

.

We say two elements $A,$$B\in\Sigma$ are equivalent if$c_{\Sigma}(A)=c_{\Sigma}(B)$. For each $A\in\Sigma$,

the subspace $U$ of $V$ occurring in Theorem 3.3 contains exactly one of the [V,$T$],

where $T$ runs through the equivalence class of $A$

.

In the case where all [V, $T$] are

1-dimensional and the equality $C_{\Sigma}(A)=c_{\Sigma}(B)$ implies that $A=B$, we see that

$U=V$ and $G=G_{0}$. Hence this assumption may replace the one that $C_{V}(G)=0$ in

Theorem 3.2 (except for the statement on the kernel of $\varphi$).

We may overcome the assumption in Theorem 3.3 that [V, $A$] is 1-dimensional as

3.4 Proposition. In the setting

_of

3.1 the subspace $R= \bigcap_{A\in\Sigma}[V, A]$ is contained in $C_{V}(G)$. Moreover, the codimension

of

$R$ in [V, $A$], $A\in\Sigma$, is one, so that we may

apply Theorem 3.3 to $G/N\leq \mathrm{G}\mathrm{L}(V/R)$, where $N$ is the kernel

of

the action

of

$G$ on

$V/R$

.

The proofs of these results are mainly geometric. They can be found in [8]. As may

be clear from the examples appearing in the conclusion of the theorems, all groups act

as automorphismgroups on a polar space. The main ideain the proof of Theorem3.2

is to construct this polar space together with an embedding into the projective space

$P(V)$ of $V$. Once this is done, we can apply the full strength of the theory of polar

spaces and its embeddings. In particular, we use the classification $\mathrm{o}\mathrm{f}\backslash$ non-degenerate

Moufang polar spaces due to Tits and the classification of their weak embeddings (of

degree $>2$) by Steinbach and Van Maldeghem [18], [21]. We find the isomorphism

type of the group $G$ in Theorem 3.2 by identifying $G$ as a group ofautomorphisms of

a polar space. The action of $G$ on the space $V$ in 3.2 is determined by the embedding

of this polar space in $P(V)$.

We note that rather than proving that the polar space constructed in Timmesfeld

[25] is weakly embedded in $P(V)$ (which is not obvious), we preferred to construct

a polar space which is automatically weakly embedded. (Hypothesis (H) of [25] is

satisfied in our setting only as long as elements of $\Sigma$ contain at least 4 elements.) The

construction does not rely on finite dimensions, commutative fields or perfect fields

in characteristic2. It is a uniform approach resulting in all different types of classical

groups.

Theorem 3.2 is an intermediate result in the proof of 3.3. To prove 3.3 we show

that there is a subspace $U$ of $V$ such that $V=U+C_{V}(G)$ and that the centers of

abstract transvection groups in $\Sigma$ which are contained in $U$ form a non-degenerate

polar space weakly embedded in $P(U)$. We are then able to identify the subgroup $G_{0}$

of $G$ generated by those elements of $\Sigma$ that have their center in $U$ as a (quasi-simple)

classical group generated by the isotropic transvection subgroups. The group $G$ is

finally identified as a split extension of $G_{0}$ by a normal subgroup which is a direct

sum of natural modules for $G_{0}$ (or equivalently, as a classical group arising from a

degenerate form).

4

$3-\mathrm{n}_{\mathrm{a}\mathrm{n}}\mathrm{S}\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$

If $\Sigma$ is a set of abstract transvection groups of $G$ with $|A|=2$, for $A\in\Sigma$, then the

set of all non-trivial elements in the members of $\Sigma$ is a set

Finite groups generated by 3-transpositions have been studied by Fischer, who

classified the finite almost simple groups generated by 3-transpositions, see [9].

Re-cently, Cuypers and Hall [6] gave a complete classification of the centerfree

3-trans-position groups containing at least 2 commuting 3-transpositions.

Among the 3-transposition groups we find various examples in which the

3-trans-positions are in fact transvections on some natural module. Indeed, the finitary

symplectic groups $\mathrm{F}\mathrm{S}\mathrm{p}(V, f)$, where (V,$f$) is a symplectic $\mathrm{G}\mathrm{F}(2)$-space, and finitary

unitary groups $\mathrm{F}\mathrm{S}\mathrm{U}(W, h)$, with $(W, h)$ a hermitian space over $\mathrm{G}\mathrm{F}(4)$, are

generated

by their isotropic

transvections

which form a class of 3-transpositions.

There are more 3-transposition groups where 3-transpositions are transvections.

The group $\mathrm{S}_{\mathrm{P}_{2n}}(2)$ contains three classes ofirreduciblesubgroups generatedby

trans-vections: the symmetric group $S_{2n+2}$ and the two different types oforthogonal groups,

$\mathrm{O}_{2n}^{+}(2)$ and $\mathrm{O}_{2n}^{-}(2)$.

The symplectic and unitary groups fit perfectly in the scheme of this paper. With

some extra effort we could have extended our methods to include these groups in

Theorem 3.2. Indeed, the set of centers of the isotropic transvections in these groups

carries the structure of a polar space, see Cuypers [7]. Reconstruction of this polar

space, $\mathrm{c}\mathrm{p}$. Cuypers [7, Section 3] and Hall [11, Section 2], and its embedding would

yield a result similar to Theorem 3.2. The symmetric andorthogonal groups, however,

do not fit into our scheme.

5

Some

historical

remarks.

The roots of the present paper can be found in the work of $\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}$ on linear

groups generated by transvections [15] and the work of Fischer on 3-transposition

groups, see [9]. Indeed, the study of linear groups generated by transvections has

been initiated by the work of $\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}$ who classified all irreducible subgroups

of $\mathrm{G}\mathrm{L}(V),$ $V$ finite-dimensional over a field $k$, generated by full linear transvection

subgroups. (A full linear transvection subgroup of $\mathrm{G}\mathrm{L}(V)$ consists of all transvections

to a fixed center and axis in $\mathrm{G}\mathrm{L}(V).)$ Soon after $\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}’ \mathrm{s}$ work Piper [16], [17],

Wagner [28] and Kantor [12] considered subgroups of finite linear groups generated

by transvections. They obtained results similar to Theorem 3.2. Where Piper’s and

Wagner’s approach was very geometric, Kantor’s work had a more group theoretic

flavor. He used the work of Fischer and generalizations thereof by Aschbacher [1],

Aschbacher and Hall [2], and Timmesfeld [22].

In more recent years $\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}’ \mathrm{s}$ work has been extended to greater classes

finite-dimensional vector spaces defined over arbitrary skew fields and Cameron and Hall

[3] also considered linear transvection groups acting (possibly reducibly) on a module

$V$ of arbitrary dimension. Related results can also be found in Cuypers [4] and

Timmesfeld [23].

Timmesfeldgeneralizedthe concept of 3-transpositions to that of abstract

transvec-tion groups. In [24], [25], he obtained a classification of the quasi-simple groups

generated by a class of abstract transvection groups, under some additional richness

assumption (which implies that the abstract transvection groups contain at least 4

elements). The cases where the abstract transvection groups contain less than 4

el-ements have been dealt with by Cuypers [5] (3 elel-ements) and by Cuypers and Hall

[6] (3-transpositions case). Timmesfeld’s results formed the basis

for.

the work ofthe

second author on subgroups of classical groups generated by long root elements, see

Steinbach [19], [20]. With regards to lineargroups generated by transvections, she

de-termines the modules $V$ (finite-dimensional over an arbitrary commutative field) for

the various groups of Timmesfeld’s classification $\dot{\mathrm{s}}\mathrm{u}\mathrm{c}\mathrm{h}$ that the abstract

transvection

groups are parts of the linear transvection subgroups of $\mathrm{G}\mathrm{L}(V)$.

Liebeck and Seitz [14] classified the closed subgroups of groups of Lie type over

algebraically closed fields which are generated by root elements.

The present approach (see [8]) combines both the work on linear groups

gener-ated by transvections as begun by $\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}$ and the geometric study of abstract

groups initiated by the work of Fischer on 3-transpositions to obtain a common

char-acterization of all the classical groups generated by isotropic transvection subgroups.

Although we do not make use of the various results quoted above, several ideas and

methods used in this paper come from this work. In particular, the proofs of our main

results are obtained by combining the group theoretic methods of Timmesfeld [25],

the more geometric approach of Cuypers [5] and the concept of weak embeddings as

in Steinbach [19].

References

[1] Aschbacher, M.:

On

_finite

groups generated by odd transpositions. I, II, $III_{J}IV$

.

Math. Z. 127 (1972), 45-56. J. Algebra 26 (1973), 451-459, 460-478,

479-491.

[2] Aschbacher, M., Hall, M. Jr.: Groups generated by a class

_of

efements

_of

order

3. J. Algebra 24 (1973), 591-612.

[3] Cameron, P.J., Hall, J.I.: Some $group\dot{s}$ generated by transvection subgroups. J.

[4] Cuypers, H.: Symplectic $geometries_{y}$ transvection groups, and modules. J. Comb.

Theory, Ser. A 65 (1994), 39-59.

[5] Cuypers, H.: The geometry of

_{k-tr.ansvection}

groups. Preprint TU Eindhoven

(1994).

[6] Cuypers, H., Hall, J.I.: The 3-transposition groups with triviaf center. J. Algebra

178 (1995), 149-193.

[7] Cuypers, H.: A characterization

_of

the symplectic and unitary 3-transposition

groups, Comm. Algebra 25 (1997), 3379-3403.

[8] Cuypers H., Steinbach, A.: Linear transvection groups and embedded polar

spaces, submitted, 1998.

[9] Fischer, B.: Finite groups generated by 3-transpositions. I. Invent. Math. 13

(1971), 232-246.

[10] Hahn, A. J., O’Meara,

0.

T.: The classical groups and $K$-theory. Grundlehren

der mathematischen Wissenschaften, 291. Springer, Berlin Heidelberg, 1989.

[11] Hall, J.I.: Geometry, graphs, 3-transpositions, and symplectic $\mathrm{F}_{2}$-transvection

groups, Proc. London Math. Soc. (Ser. 3) 58 (1989), 89-111.

[12] Kantor, W. M.: Subgroups

_of

classical groups generated by long root elements.

Trans. Am. Math. Soc. 248 (1979), 347-379.

[13] Li, Shangzhi: Irreducible subgroups

_of

$SL(n, I\backslash ^{\nearrow})$ generated by root subgroups.

Geom. Dedicata 31 (1989), 41-44.

[14] Liebeck, M. W., Seitz, G. M.: Subgroups generated by root elements in groups

_of

Lie type. Ann. Math., II. Ser. 139 (1994), 293-361.

[15] McLaughlin, J.: Some groups generated by transvections. Arch. der Math. 18

(1967), 364-368.

[16] Piper, F.C.: On elations

_offinite

projective spaces

_of

odd order. J. Lond. Math.

Soc. 41 (1966), 641-648.

[17] Piper, F.C.: On elations

_of

_finite

projective spaces

_of

even order. J. Lond. Math.

[18] Steinbach, A.: Classicalpolar spaces $(sub-)weakly$ embedded in projective spaces.

Bull. Belg. Math. Soc. Simon Stevin 3 (1996),

477-490.

[19] Steinbach, A.: Subgroups

_of

classical groups generated by transvections or Siegel

transvections I: Embeddings in linear groups. Geom. Ded. 68 (1997), 281-322.

[20] Steinbach, A.: Subgroups

_of

classical groups generated by transvections or Siegel

transvections II.. Embeddings in orthogonal groups. Geom. Ded. 68 (1997),

323-357.

[21] Steinbach, A., Van Maldeghem, H.: Generalized quadrangles weakly embedded

_of

degree $>2$ in projective space. To appear in Forum Math.

[22] Timmesfeld, F.G.: Groups generated by root-involutions. I. J. Algebra 33 (1975),

75-134.

[23] Timmesfeld, F.G.: On the

_{identification}

_of

natural modules

_for

symplectic and

linear groups

_defined

over arbitrary

_fields.

Geom. Dedicata 35 (1990),

127-142.

[24] Timmesfeld, F.G.: Groups generated by $k$-transvections. Invent. Math. 100

(1990), 167-206.

[25] Timmesfeld, F. G.: Abstract root subgroups and quadratic action. To appear in

Adv. Math.

[26] Tits, J.: Buildings

_of

spherical type and

_finite

$BN$-pairs, Lecture Notes in Math.

386. Springer-Verlag, Berlin Heidelberg New York, 1974.

[27] Vavilov, N.A.: Linear groups generated by one-parameter groups

_of

one-dimensional

_{transformations.}

Russ. Math. Surv. 44 (1989), 265-266.

[28] Wagner, A.: Groups generated by elations. Abh. Math. Semin. Univ. Hamb. 41

(1974), 190-205.

Hans Cuypers Anja Steinbach

Department of Mathematics Mathematisches Institut

Eindhoven University of Technol- Justus-Liebig-Universit\"at $\mathrm{G}\mathrm{i}\mathrm{e}\mathfrak{g}\mathrm{e}\mathrm{n}$

ogy Arndtstralie 2

$\mathrm{P}.\mathrm{O}$

.

BOX 513 $\mathrm{D}$ 35392 $\mathrm{G}\mathrm{i}\mathrm{e}\mathfrak{g}\mathrm{e}\mathrm{n}$

5600 MB Eindhoven Germany

The Netherlands email: [email protected]